AAS 16-495

NONLINEAR ANALYTICAL EQUATIONS OF RELATIVE MOTIONON J2-PERTURBED ECCENTRIC ORBITS

Bradley Kuiack∗ and Steve Ulrich†

Future spacecraft formation flying missions will require accurate autonomous guid-ance systems to calculate a reference, or desired, relative motion during reconfig-uration maneuvers. However, the efficiency in terms of propellant used for suchmaneuvers depends on the fidelity of the dynamics model used for calculatingthe reference relative motion. Therefore, an efficient method for calculating rela-tive motion should have an analytical solution, be applicable to an eccentric orbit,and should take into account the J2 perturbation. This paper accomplishes thisthrough an exact analytical solution of the relative motion between two spacecraftbased on the orbital elements of each spacecraft. Specifically, by propagating theJ2-perturbed osculating orbital elements forward in time and solving the exact so-lution at each time step, an accurate representation of the true spacecraft relativemotion is obtained. When compared to a numerical simulator, the proposed an-alytical solution is shown to accurately model the relative motion, with boundederrors on the order of meters over a wide range of eccentricity values.

INTRODUCTION

Formation flying of multiple spacecraft is a key enabling technology for many future scientificmissions such as enhanced stellar optical interferometers and virtual platforms for Earth or Sunobservations. However, formation flying involves several considerations beyond those of a sin-gle spacecraft mission. This is particularly true for the guidance system, which is responsible forcalculating a reference, or desired, relative motion to be actively tracked during reconfiguration ma-neuvers by the on-board control system of a follower spacecraft. Such guidance systems requireaccurate but simple dynamics models of the spacecraft relative motion. Specifically, the dynamicsmodels have to be accurate enough to prevent unnecessary fuel expenditure, as ignoring perturba-tions in the design of the reference relative trajectories will most likely result in additional propellantconsumption to force the follower spacecraft to track artificial reference trajectories. Indeed, to limitthe propellant consumption, the follower spacecraft must avoid compensating for orbital perturba-tions, as well as other nonlinear effects inherent to large separation distances and/or eccentricityvalues. In other words, the effectiveness of a reference relative motion depends on the accuracy ofthe dynamics model used for its design. In addition, the guidance system must remain computation-ally simple for on-board implementation purposes. As such, analytical solutions are preferable asthey do not rely on numerical integrations.

∗Undergraduate Student, Department of Mechanical and Aerospace Engineering, Carleton University, 1125 Colonel ByDrive, Ottawa, ON, K1S 5B6, Canada.†Assistant Professor, Department of Mechanical and Aerospace Engineering, Carleton University, 1125 Colonel By Drive,Ottawa, ON, K1S 5B6, Canada.

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While some analytical solutions to the exact nonlinear differential equations of relative motionin the local-vertical-local-horizontal reference frame do exist, they nevertheless impose certain re-strictions in their derivations. For example, the Hill-Clohessy-Wiltshire (HCW) model,1 does havea time-explicit closed-form analytical solution to the relative motion problem, but only for circularKeplerian orbits. Sabol and McLaughlin2 reparametrized this model and demonstrated that the in-plane and out-of-plane non-drifting relative motion about a circular Keplerian orbit always followsan ellipse centered on the reference orbit.

To overcome the inherent limitations of the HCW model, some formulations also take into ac-count orbit perturbations, notably the perturbation caused by the oblateness of the Earth, referredto as the J2 perturbation. The J2 perturbation is particularly important in formation flying since itssecular effect on an orbit, i.e., the rotation of the line of apsides and precession of the line of nodes,causes secular drift between two J2-perturbed spacecraft. The introduction of J2 in a linearizedset of equations, similar to the HCW equations, has been proposed by Schweighart and Sedwick.3

Specifically, the authors developed a set of linear, constant-coefficient, second-order differentialequations of motion by considering the orbit-averaged impact of J2 on a circular reference orbit.An analytical solution to these differential equations is also presented.

However, assuming a circular reference orbit yields considerable errors when the eccentricity ofthe reference orbit increases. In fact, it has been shown that the errors introduced in the HCW equa-tions by considering an elliptical reference orbit dominate the errors due to ignoring J2.4 For thisreason, several formulations have been proposed to model the relative motion about unperturbed el-liptical orbits.5–7 In particular, the linear-time-invariant (LTI) HCW equations have been extended toarbitrary eccentricity by Inalhan et al.4 by formulating the dynamics as a linear-parameter-varying(LPV) system. Such a dynamics model is especially well suited for controller design purposes,by making use of LPV control techniques, such as model predictive control.8 Making use of thefact that the orbit angular rate and the radius are functions of the true anomaly, these equations canalso be expressed using true anomaly derivatives instead of time derivatives.9 The resulting dif-ferential equations are also time-varying, but are not parameter-varying since the true anomaly hasbeen used to formulate the derivatives. Solutions to these linear-time-varying (LTV) equations areavailable in the literature in various forms using different reference frames and variables. The firstderivation with singularities in the closed-form solution was provided by Lawden.10 Then, Carter11

provided a set of solutions without singularities. These homogenous solutions are extremely use-ful for well-behaved numerical and analytical analysis on the shape, structure, and optimization ofpassive apertures in eccentric reference orbits. Interestingly, the same solutions can be obtainedvia incremental changes in orbital elements, as demonstrated by Marec.12 Lane and Axelrad13 de-veloped a time-explicit closed-form solution and studied the relative motion for bounded ellipticalorbits. Melton14 also proposed an alternative solution for small eccentricity reference orbits. Morerecently, an elegant closed-form analytical solution that is valid for Keplerian eccentric orbits wasdeveloped by Gurfil and Kholshevnikov.15, 16 This simple analytical solution explicitly parameter-izes the relative motion using classical orbital elements as constants of the unperturbed Keplerianorbit, instead of Cartesian initial conditions. This concept, originally suggested by Hill,17 has beenwidely used in the analysis of relative spacecraft dynamics.18, 19 A key advantage of this approachis that it facilitates the inclusion of the orbital perturbation effects on the relative motion.

In this context, the original contribution of this paper is to propose an improvement to Gurfiland Kholshevnikov’s equations of relative motion15, 16 by taking into account J2 perturbation. Thisis achieved by using J2-perturbed osculating orbital elements in their solutions instead of using

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constant orbital elements. The osculating orbital elements are calculated by adding the mean orbitalelements to the J2-induced short-periodic variations.20, 21 In addition, the original solution by Gurfiland Kholshevnikov was not fully analytical as numerical integrations were employed to calculatethe true anomaly of the leader spacecraft and the eccentric anomaly of the follower spacecraft.15, 16

This problem is herein alleviated by using an analytical approximation of the eccentric anomalies,which is then used to calculate the true anomalies of both spacecraft.

NONLINEAR ANALYTICAL EQUATIONS OF RELATIVE MOTION

This section details the methods used to derive a set of nonlinear analytical equations of relativemotion which are applicable to J2 perturbed eccentric orbits.

Reference Frames

For completeness, the reference frames that will be subsequently used in the derivations of theequations of motion are first introduced.

The first is the Earth-centered inertial (ECI) reference frame, denoted by FI and defined by itsorthonormal unit vectors

~Ix, ~Iy, ~Iz

. This reference frame has its origin at the center of the Earth,

its ~Iz is along the Earth’s polar axis of rotation, and its ~Ix and ~Iy unit vectors in the Earth’s equatorialplane. Specifically, the unit vector ~Ix of ECI is along the vernal equinox line in the direction of thevernal equinox (sometimes called the first point of Aries), and ~Iy = ~Iz × ~Ix.

The perifocal frame, denoted by FP and defined by its orthonormal unit vectors~Px, ~Py, ~Pz

,

is also centered at the center of the Earth, but with its ~Pz aligned with a spacecraft’s orbit normalvector, its ~Px pointing in the direction of the perigee, and ~Py = ~Pz × ~Px. This frame is specific tothe orbit of a given spacecraft. As a result, two seperate perifocal frames will be used in this paper,i.e., one for each spacecraft. These frames will be denoted FP and FP ′ for the perifocal frame ofthe follower and the leader spacecraft, respectively. The corresponding unit vectors will be labeled~Px, ~Py, ~Pz

and

~P′x,~P′y,~P′z

for the follower and leader spacecraft, respectively.

The last reference frame relevant to this work is the local-vertical-local-horizontal (LVLH) refer-ence frame denoted byFL and defined by its orthonormal unit vectors

~Lx, ~Ly, ~Lz

. This reference

frame has its origin at the leader spacecraft, with its ~Lx unit vector aligned with the spacecraft po-sition vector and its ~Lz unit vector aligned with the orbit normal. The ~Iy unit vector completes theright-handed triad. Note that in the case of a perfectly circular orbit, ~Iy is in the direction of the ve-locity vector. The solution to the equations of motion developed in this paper provides an analyticalexpression of the position of the follower spacecraft with respect to the leader in FL.

Relative Motion on Keplerian Eccentric Orbits

As previously mentioned, previous work by Gurfil and Kholshevnikov15, 16 form a basis for theanalytical solution derived herein. In their work, Gurfil and Kholshevnikov proposed a simplemethod of calculating the relative position of a follower spacecraft with respect to a leader usingorbital elements. This allows the equations of relative motion to remain accurate for arbitrarilylarge values of eccentricity. To do so, consecutive rotations and a translation are applied to obtainan analytical expression for ρ, defined as the components of the relative position vector in FL, thatis

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ρ = rL − r′L (1)

where rL and r′L respectively denote the components of the follower and leader spacecraft positionvector expressed inFL. The first step in calculating ρ is to transform the components of the followerposition vector from FP to FI using CIP (ω, i,Ω), the 3-1-3 rotation sequence given by

CIP (ω, i,Ω) =

cΩcω − sΩsωci −cΩsω − sΩcωci sΩsisΩcω + cΩsωci −sΩsω − cΩcωci −cΩsi

sωsi cωsi ci

(2)

where Ω, i, and ω respectively denote the right ascension of the ascending node (RAAN), inclina-tion, and argument of perigee of the follower spacecraft. The next step is to transform the compo-nents of the follower position vector from FI to FP ′ using CP ′I(ω

′, i′,Ω′) given by

CP ′I(ω′, i′,Ω′) =

cΩ′cω′ − sΩ′sω′ci′ sΩ′cω′ + cΩ′sω′ci′ sω′si′

−cΩ′sω′ − sΩ′cω′ci′ −sΩ′sω′ − cΩ′cω′ci′ cω′si′

sΩ′si′ −cΩ′si′ ci′

(3)

where Ω′, i′, and ω′ denote the leader’s RAAN, inclination, and argument of perigee, respectively.Then, the components of the follower position vector are rotated from FP ′ to FL with CLP ′ (ν

′)

CLP ′ (ν′) =

cν′ sν′ 0−sν′ cν′ 0

0 0 1

(4)

where ν ′ denotes the true anomaly of the leader spaceraft. Finally, a translation is applied, such that

ρ = CLP ′ (v′)CP ′I(ω

′, i′,Ω′)CIP (ω, i,Ω)rP − r′L (5)

where rP and r′L represents the components of the follower’s position vector in FP , and the com-ponents of the leader’s position vector in FP , respectively given by

rP =[r cos ν r sin ν 0

]T (6)

r′L =[r′ 0 0

]T (7)

where v denotes the true anomaly of the follower spaceraft, and where the orbit radii, r and r′, areobtained from the orbit equation

r =a(1− e2)

1 + e cos ν(8)

r′ =a′(1− e′2)

1 + e′ cos ν ′(9)

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with a and e denoting the semimajor axis and eccentricity of the follower, and a′ and e′ denotingthe semimajor axis and eccentricity of the leader spacecraft. Note that Eq. (5) can be writtencomponent-wise and further simplified through the use of relative orbital elements. For more details,the readers are referred Refs.15, 16

Relative Motion on J2-Perturbed Eccentric Orbits

The previous equations use constant orbital elements to calculate the position of a follower space-craft in FL. Using constant orbital elements is a good approximation for the solution, and is de-sirable as it greatly simplifies the computations involved. The effectivenes of these equations wasvalidated at a variety of eccentricity values, ranging from 0 to 0.8, proving that the equations arequite effective at predicting relative motion in eccentric Keplerian orbits (not shown here, for con-ciseness). However, perturbing accelerations will have a significant effect on the relative motion ofthe spacecraft. The most dominant, and therefore the most important to consider, perturbation inlow-Earth orbits is the J2 gravitational perturbation. One notable consequence of J2 on the relativemotion is a drift between both spacecraft along the ~Ly direction. Figure 1 shows an example of thismotion for a typical satellite formation.

Figure 1. Simulated Relative Motion of Two Spacecraft with and without J2 Perturbation

It should be noted, however, that Eq. (5) represents an exact solution for the relative positionof the follower spacecraft at any given time, assuming that the orbital elements for each spacecraftused in the equation are accurate. Therefore, if accurate J2-perturbed osculating orbital elementsare used instead of constant ones, Eq. (5) would then result in an accurate solution for the relativeposition between both spacecraft under J2. In other words, J2-perturbed osculating orbital elementsfor each spacecraft can be used with Eq. (5) to accurately predict ρ.

The osculating orbital elements, [a, e, i, ω,Ω,M ]T , can be accurately predicted by summingmean orbital elements with the J2-induced short-periodic variations, as follows

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a = a+ ∆asp

e = e+ ∆esp

i = i+ ∆isp

ω = ω + ∆ωsp

Ω = Ω + ∆Ωsp

M = M + ∆Msp

(10)

where [∆asp,∆esp,∆isp,∆ωsp,∆Ωsp,∆Msp]T , denote the short-periodic variations of the orbital

elements, and where the mean orbital elements,[a, e, i, ω, Ω, M

]T , can be described as the initialmean orbital elements, denoted by

[a0, e0, i0, ω0, Ω0, M0

]T , summed with the secular variationsdue to J2

a = a0

e = e0

i = i0

ω = ω0 + ˙ωt

Ω = Ω0 + ˙Ωt

M = M0 + ˙Mt

(11)

where t is the elasped time since t0. It has been shown that the orbital elements a, e, and i havezero secular change, while the remaining elements do have some nonzero secular rate of change,

denoted by[

˙ω, ˙Ω, ˙M]T

, as a result of the J2 perturbation21

˙ω =1

2nJ2

(Rep

)2(4− 5 sin2 i) (12)

˙Ω = −3

2nJ2

(Rep

)2cos i (13)

˙M = n+3

2nJ2

(Rea

)2 1

(1− e2)3/2

(1− 3

2sin2 i

)(14)

where Re is the mean equatorial radius of the Earth, and where the mean orbital motion and semi-latus rectum calculated with the mean semimajor axis and eccentricity, denoted as n and p, respec-tively, are given by

n =

õ

a3(15)

p = a(1− e2) (16)

Now that the mean orbital elements are fully defined for each spacecraft, the short periodic vari-ations of each element must be calculated, as follows20

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∆asp =J2R

2e

a

[(ar

)3− 1

(1− e2)3/2+−(ar

)3+

1

(1− e2)3/2+(ar

)3cos(2ω + 2ν)

3 sin2 i

2

](17)

∆esp =J2R

2e

4

[ −2

a2e√

1− e2+

2a(1− e2)

er3+ 3

a2e√

1− e2− 3a(1− e2)

er3− 3(1− e2) cos(ν + 2ω)

p2

− 3 cos(2ν + 2ω)

a2e(1− e2)+

3a(1− e2) cos(2ν + 2ω)

er3− (1− e2) cos(3ν + 2ω)

p2

sin2 i

](18)

∆isp =J2R

2e sin(2i)

8p2[3 cos(2ω + 2ν) + 3e cos(2ω + ν) + e cos(2ω + 3ν)] (19)

∆Ωsp =J2R

2e cos(i)

4p2[6(ν−M+e sin ν)−3 sin(2ω+2ν)−3e sin(2ω+ν)−e sin(2ω+3ν)] (20)

∆ωsp =3J2R

2e

2p2

[(2− 5

2sin2 i

)(ν −M + e sin ν)

+(

1− 3

2sin2 i

)1

e

(1− 1

4e2)

sin ν +1

2sin(2ν) +

e

12sin(3ν)

− 1

e

1

4sin2 i+

(1

2− 15

16sin2 i

)e2

sin(ν + 2ω) +e

16sin2 i sin(ν − 2ω)

− 1

2

(1− 5

2sin2 i

)sin(2ν + 2ω) +

1

e

7

12sin2 i− 1

6

(1− 19

8sin2 i

)e2

sin(3ν + 2ω)

+3

8sin2 i sin(4ν + 2ω) +

e

16sin2 i sin(5ν + 2ω)

](21)

∆Msp =3J2R

2e

√1− e2

2ep2

[−(

1− 3

2sin2(i)

)(1− e2

4

)sin ν +

e

2sin(2ν) +

e2

12sin(3ν)

+ sin2 i

1

4

(1 +

5

4e2)

sin(ν + 2ω)− e2

16sin(ν − 2ω)− 7

12

(1− e2

28

)sin(3ν + 2ω)

− 3e

8sin(4ν + 2ω)− e2

16sin(5ν + 2ω)

](22)

Even though ν is not explicitly part of the orbital elements, its accurate evaluation is neverthelessrequired to make use of the previously defined equations. This calculation must obviously accountfor the J2 perturbation, as the perturbation will have an effect on the true anomaly. In their paper,Gurfil and Kholshevnikov proposed to numerically integrate an expression for the time derivative ofthe true anomaly. However, doing so is undesirable. Fortunately, ν can be obtained iteratively fromthe mean anomaly and the orbit eccentricity through the well-known relations

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E = M + e sin(M + e sin(M + e sin(M + · · ·+ e sin(M)))) (23)

cos ν =cosE − e

1− e cosE(24)

sin ν =

√1− e2 sinE

1− e cosE(25)

ν = tan−1 sin ν

cos ν(26)

where E denotes the eccentric anomaly. The number of terms in Eq. (23) is selected in accordancewith the desired accuracy and is strictly a function of the magnitude of the eccentricity.

PROCEDURE FOR FINDING RELATIVE MOTION SOLUTION

In the previous section, all of the necessary equations were presented for solving analyticallyfor the relative position between two spacecraft, i.e., by using Eq. (5) with J2-perturbed osculatingorbital elements. The purpose of this section is to summarize the steps taken to apply these equationsto propagate the relative position vector.

1. Two sets of Keplerian initial osculating orbital elements are specified, i.e., [a0, e0, i0, ω0,Ω0, ν0]T

and [a′0, e′0, i′0, ω′0,Ω

′0, ν′0]T .

2. The mean anomaly of each spacecraft most also be initialized, i.e., M0 and M ′0. This isaccomplished by working backwards from the true anomaly to the eccentric anomaly andfinally mean anomaly using Eqs. (23)-(26).

3. The initial short-periodic variations for each spacecraft are calculated as a function of theinitial osculating orbital elements using Eqs. (17)-(22).

4. By rearranging Eq. (10), the initial mean orbital elements for each spacecraft can then be cal-culated from the initial short-periodic variations subtracted from the initial osculating orbitalelements, that is

a0 = a0 −∆a0sp

e0 = e0 −∆e0sp

i0 = i0 −∆i0sp

ω0 = ω0 −∆ω0sp

Ω0 = Ω0 −∆Ω0sp

M0 = M0 −∆M0sp

and similarly for the leader spacecraft.

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5. The mean semimajor axis, eccentricity and inclination have no secular variation, and aretherefore equal to their initial mean values, as calculated in the previous step, i.e., a = a0,e = e0, and i = i0, as well as a′ = a′0, e′ = e′0, and i′ = i′0. The mean orbital motion andmean semilatus rectum for each spacecraft, n, p, n′, and p′, may also be calculated at thispoint using Eqs. (15) and (16), as these parameters are only functions of mean semimajoraxis and mean eccentricity.

6. The remaining mean orbital elements,[ω, Ω, M

]T and[ω′, Ω′, M ′

]T , can be propagated for-ward in time, from t0 to t, using Eqs. (12)-(14) with Eq. (11).

7. Based on the average mean anomalies, M and M ′, the true anomaly for each spacecraft, νand ν ′, are propagated forward in time using Eqs. (23)-(26).

8. Now that the true anomalies have been propagated forward in time, the short-periodic varia-tions can then be calculated for each spacecraft using Eqs. (17)-(22). The osculating orbitalelements are then propagated forward in time using Eq. (10).

9. Finally, ρ is obtained through Eq. (5), with rotation matrices in Eqs. (2)-(4) and positionvectors in Eqs. (6) and (7) evaluated with the J2-perturbed osculating orbital elements.

NUMERICAL SIMULATIONS

To verify the accuracy of the newly developed method of calculating the relative position betweentwo spacecraft, numerical simulations were performed in MATLAB. The results obtained with theanalytical solution provided in this paper were compared to a numerical simulator that integratesthe exact, nonlinear differential equations of motion in FI . Specifically, the simulator integrates,for each spacecraft, the inertial two-body equations of motion to which the J2 inertial perturbing ac-celeration was added. Both spacecraft’s position vectors are then transformed into FL. Finally, theleader position vector is subtracted from the follower position vector, to obtain ρ, the componentsof the exact relative position vector in FL. The simulator was initialized using, for both spacecraft,the initial osculating orbital elements which were converted into components of the initial positionand velocity vectors in FI . The exact same initial osculating orbital elements were also used to ini-tialize the proposed method of calculating relative position outlined in the previous section. Unlessotherwise specified in the captions of the subsequent figures, Table 1 presents the initial osculatingorbital elements used for both models (numerical simulator and analytical equations). The orbitalelements given in this Table correspond to a simple in-plane elliptical formation.

Table 1. Initial Osculating Orbital Elements

Orbital Element Leader Follower

Semimajor Axis (km) 7106.14 7106.14Eccentricity 0.05 e′ + 0.001Inclination (deg) 98.3 98.3Argument of Perigee (deg) 0 0Right Ascension of Ascending Node (deg) 270 270True Anomaly (deg) 0 0

Figure 2 reports the results obtained by propagating the relative motion at a relatively low ec-centricity over six orbits with the analytical equations and the numerical simulator. As shown, theposition error, defined as difference between the analytical solution and the numerical simulator,

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remains bounded (for all practical purposes) with an accuracy better than 5 meters along each di-rection. While there is a slight increase in the error in the along-track direction, this minimal driftis negligible over the six orbital periods shown.

Figures 3 and 4 show the results obtained when the leader reference orbit is significantly more el-liptical. Similarly to the previous near-circular orbit case, the errors are bounded, and the analyticalsolution is representative of the exact spacecraft motion. However, it is clear that as the eccentricityincreases the ability of the analytical solution to track the relative motion accurately decreases. Thisis especially true in the along-track direction, where the error increases significantly with increasein eccentricity.

Proving that the new analytical solution can predict relative postion in highly eccentric orbits isparticularly important. To this end, a simulation was performed with e′ = 0.8. Figure 5 shows thatthe analytical equations do begin to show unbounded error at such high eccentricity values, withlarge error spikes reaching several kilometers after six orbits. Looking at the left side of Figure 5 itcan be seen that the analytical solution still models the simulated motion well despite the increasingerror, and it can also be seen that over time, as the satellites drift apart due to J2, the errors get worse.It should be noted that accuracy of Eq. (23) is a function of the number of iterations performed in theequation. When the eccentricity was increased to 0.8, the resulting errors were initially significantlylarger. By increasing the iterations in Eq. (23) this error was reduced to the results shown in Figure5. To further study the application of the proposed analytical solution for formations on highly-eccentric orbits, the relative equations of motion were applied to the upcoming European SpaceAgency PROBA-3 mission. This Sun observation mission will involve two spacecraft in preciseformation in highly-eccentric orbits, with the formation forming a solar coronagraph. The PROBA-3 leader’s orbit is defined with a′0 = 37040 km, e′0 = 0.806, i′0 = 59 deg, Ω′0 = 84 deg, and ω′0 = 188deg, as well as a spacecraft speration on the order of 100 meters.22 Setting up a simulation withtwo spacecraft in this orbit, with the follower eccentricity given by e = e′ + 0.00005 is a grosssimplification of the ultimate relative motion for the mission, but it may still provide some insightregarding the accuracy of the proposed analytical solution in a realistic mission scenario. Figure6 shows the results of the simulation. Although the resulting errors are unbounded, the error iscontrolled quite well, staying below 40 meters over six orbital periods.

In each of the previous cases, the formation was defined such that there is no out-of-plane motion.As it is important to assess the ability of the analytical solution to handle out-of-plane motioneffectively, three more cases were simulated at varying inclination differences between the twospacecraft. The results, in order of increasing inclination difference are reported in Figures 7, 8, and9.

CONCLUSION

A set of nonlinear analytical equations for the relative motion between two spacecraft in forma-tion which is applicable to a wide variety of orbits, and also takes into account the J2 perturbationwas derived in this paper. By building upon previously-published results, the relative motion is for-mulated as a function of osculating J2-perturbed orbital elements, such that an analytical solutionthat is exact for a perturbed orbit of any shape was used. This new method of obtaining the rela-tive position between two spacecraft in formation was validated in numerical simulations. Whencompared to a numerical simulator, the analytical solution follows the simulated motion accuratelyalong all three directions, and the relatively small errors are bounded for most cases. Future workwill focus on reformulating the proposed analytical solution into component-wise equations. This

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Figure 2. Analytical Solution Compared with Numerical Simulator

could potentially decrease the computational resources required to use the proposed analytical so-lution in an on-board guidance system. Additionally, in practice, it is likely not desired to have twospacecraft drifting apart due to the J2 perturbation. Further examination of the analytical equationsin a component-wise form would allow the elimination of this drift by careful selection of the initialosculating orbital elements. This would be analogous to establishing the formation in a widely-usedJ2-invariant orbit.

ACKNOWLEDGMENTS

This work was financially supported by the Natural Sciences and Engineering Research Councilof Canada under the Undergraduate Student Research Award Program and Discovery Grant Pro-gram. Special thanks to John and Lorna Kuiack for all of their support, and Giuliana Velarde for allof her editing help.

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Figure 3. Analytical Solution Compared with Numerical Simulator for e′ = 0.2 and a′ = 9, 000 km

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Figure 4. Analytical Solution Compared with Numerical Simulator for e′ = 0.4 and a′ = 12, 000 km

Figure 5. Analytical Solution Compared with Numerical Simulator for e′ = 0.8 and a′ = 34, 000 km

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Figure 6. Analytical Solution Compared with Numerical Simulator for PROBA-3 example

Figure 7. Analytical Solution Compared with Numerical Simulator for i = i′ + 0.001 deg

Figure 8. Analytical Solution Compared with Numerical Simulator for i = i′ + 0.002 deg

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Figure 9. Analytical Solution Compared with Numerical Simulator for i = i′ + 0.004 deg

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